How the two-way table fits together
A two-way ANOVA with interaction crosses factor A (a levels) with factor B (b levels), both fixed, with n observations in each of the ab cells, N = abn in all. The total spread splits four ways: factor A, factor B, the interaction and what is left within the cells.
| Source | SS | df | MS | F |
|---|---|---|---|---|
| Factor A | SS A | a − 1 | SS A ÷ df A | MS A ÷ MS within |
| Factor B | SS B | b − 1 | SS B ÷ df B | MS B ÷ MS within |
| Interaction | SS interaction | (a − 1)(b − 1) | SS interaction ÷ its df | MS interaction ÷ MS within |
| Within | SS within | ab(n − 1) | SS within ÷ df within | |
| Total | Sum of the four | N − 1 |
The SS column and the df column both add up to their totals. With both factors fixed, every F has the same denominator, MS within, and each is compared with the critical value for its own df₁ and df₂ = df within. Read the interaction first: if it is significant, the effect of A depends on the level of B.
An example from set EXD32K
A study crosses 2 study methods with 3 amounts of sleep, 6 students in each of the 6 cells, so N = 36. The df are 1, 2, 2 (that is 1 × 2), 30 and 35. The table gives SS sleep = 647.64, SS interaction = 1,181.88, SS within = 1,890 and SS total = 3,913.56, so SS study method = 3,913.56 − 647.64 − 1,181.88 − 1,890 = 194.04. MS within = 1,890 ÷ 30 = 63, and the three F values are 194.04 ÷ 63 = 3.08, 323.82 ÷ 63 = 5.14 and 590.94 ÷ 63 = 9.38. At α = 0.05, F(1, 30) = 4.17 and F(2, 30) = 3.32: sleep and the interaction are significant, study method is not. Open set EXD32K
Questions
What are the degrees of freedom in a two-way ANOVA?
For factor A with a levels, factor B with b levels and n observations in each of the ab cells: df A = a − 1, df B = b − 1, df interaction = (a − 1)(b − 1), df within = ab(n − 1) = N − ab and df total = N − 1. With a = 2, b = 3 and n = 6 that is 1, 2, 2, 30 and 35, which add up to 35.
How do you calculate F in a two-way ANOVA?
Each effect has its own F: its MS divided by MS within (also called MS error). So F for A = MS A ÷ MS within, F for B = MS B ÷ MS within and F for the interaction = MS interaction ÷ MS within. All three use the same denominator.
Which critical value do you use for each F?
Read each one from the F table with df₁ = that row's df and df₂ = df within. Two effects with the same df share a critical value. With df within = 30 at α = 0.05, df₁ = 1 gives 4.17 and df₁ = 2 gives 3.32.
What does a significant interaction mean?
That there is evidence the effect of one factor depends on the level of the other. When the interaction is significant, compare the cell means (an interaction plot helps) before saying anything about either main effect on its own.
Why is the interaction df a product and not a sum?
The interaction is what is left in the ab cell means after both main effects are taken out: ab − 1 − (a − 1) − (b − 1) = (a − 1)(b − 1). Adding the two df instead is a common slip.
Sources, checked 2 November 2026, all from the NIST/SEMATECH e-Handbook of Statistical Methods: 7.4.3.7 The two-way ANOVA, 7.4.3.8 Models and calculations for the two-way ANOVA and 1.3.6.7.3 Upper critical values of the F distribution.